D. RICO MUNN (D) 1st Congressional District

CLAIR ORR (R) 4th Congressional District

September 2005 Colorado Department of Education

Colorado Model Content Standards

MATHEMATICS

Standard 1

Students develop number sense and use numbers and number relationships in problem-solvingsituations and communicate the reasoning used in solving these problems.

Standard 2

Students use algebraic methods to explore, model, and describe patterns and functions involvingnumbers, shapes, data, and graphs in problem-solving situations and communicate the reasoningused in solving these problems.

Standard 3

Students use data collection and analysis, statistics, and probability in problem-solving situationsand communicate the reasoning used in solving these problems.

Standard 4

Students use geometric concepts, properties, and relationships in problem-solving situations andcommunicate the reasoning used in solving these problems.

Standard 5

Students use a variety of tools and techniques to measure, apply the results in problem-solvingsituations, and communicate the reasoning used in solving these problems.

Standard 6

Students link concepts and procedures as they develop and use computational techniques,including estimation, mental arithmetic, paper-and-pencil, calculators, and computers, in problem-solving situations and communicate the reasoning used in solving these problems.

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Colorado Model Content Standards for Mathematics

HISTORY

The Colorado Model Content Standards for Mathematics were developed by Colorado math teachers and adopted by the Colorado State Board of Education on June 8, 1995. Three questions guided the development of these standards:

• What is mathematics? • What does it mean to know, use, and do mathematics? • What mathematics should every Colorado student learn?

The Colorado Model Content Standards for Mathematics were reviewed by the Colorado Department of Education during the 2004-2005 school year concluding with the report titled The State’s Prime Numbers. No changes to the Colorado Model Content Standards for Mathematics were recommended.

The Colorado State Board of Education reaffirmed the Colorado Model Content Standards for Mathematics on September 14, 2005 and amended the Benchmarks to Standard 6. Changes to the benchmarks are noted in bold/capitalized type below.

Students link concepts and procedures as they develop and use computational techniques,Standard 6 including estimation, mental arithmetic, paper-and-pencil, calculators, and computers, in problem-solving situations and communicate the reasoning used in solving these problems.

addition, subtraction, multiplication, and division facts without the use of a calculator.

Benchmark 5 Select and use appropriate methods ALGORITHMS for computing with whole numbers in problem-solving situations from among mental arithmetic, estimation, paper-and-pencil, calculator, and computer methods.

Grades 5 - 8 Benchmark 4 Select and use appropriate methods ALGORITHMS for computing with commonly used fractions and decimals, percents, and integers in problem-solving situations from among mental arithmetic, estimation, paper-and-pencil, calculator, and computer methods and determining whether the results are reasonable.

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HISTORY

Standards and Benchmarks

The state of Colorado's education system is operated locally. There are state standards and the commensurate benchmarks and assessment frameworks which articulate more specific areas of focus expected at grade levels. The annual state assessment is administered grades 3-10 in reading, writing, and math. In 2006, the state assessment will include 5th, 8th and 10th grade science. CSAP Assessment Frameworks exist for these specific areas.

The Colorado Model Content Standards for Mathematics indicate the broad knowledge and skills that all students should acquire in Colorado schools. In this document, standards are articulated into benchmarks that include tactical descriptions of the knowledge and skills students should acquire within each grade level range.

CSAP and Assessment Objectives

The Assessment Frameworks for the Colorado Student Assessment Program (CSAP) outlines what is assessed on the state paper and pencil, standardized, and timed assessment. Assessment objectives delineate the specific knowledge and skills measured by CSAP for each grade level and content area assessed. The CSAP Assessment Frameworks are available on the Colorado Department of Education website (http://www.cde.state.co.us).

Curriculum and Instructional Objectives

Colorado has no state curriculum. Local school districts in Colorado are responsible for determining the necessary curriculum and instructional scope and sequence to ensure that their students meet state standards.

The Colorado Department of Education provides a “resource bank” of curriculum, instruction

and assessment tools acquired from Colorado schools that are achieving positive results in mathematics to be used by school districts at their discretion. The Colorado Math webpage provides: resources to address the needs of students performing at grade level, as well as struggling and advanced learners; model programs of instruction and assessment collected from school districts and organizations throughout the state and nation that have proven to be successful; and many resources that may assist Colorado’s mathematics educators in enhancing their teaching methods and improving student performance outcomes (http://www.cde.state.co.us/coloradomath/index.htm).

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Standard 1:Students develop number sense and use numbers and number relationships inproblem-solving situations and communicate the reasoning used in solving theseproblems.

RATIONALENumbers play a vital role in our daily lives. It is essential to know both the symbols for and the meanings of various kindsof numbers; whole numbers, fractions, decimals, percents, roots, exponents, logarithms, and scientific notation. Numbersense is the capacity a child has to be flexible and mentally agile with numbers; to have a working knowledge for whatnumbers mean and an ability to perform mental mathematics.Number sense enables a student to look at the world through the eyes of math and make comparisons and build newinformation (Case 1998). Developing number sense strengthens students' ability to acquire basic facts, to solveproblems, and to determine the reasonableness of results.BENCHMARKSGRADES K-4 GRADES 9-12

1. demonstrate meanings for whole numbers, and commonly- 1. demonstrate meanings for real numbers, absolute value, and used fractions and decimals (for example, 1/3, 3/4, 0.5, scientific notation using physical materials and technology in 0.75), and represent equivalent forms of the same number problem-solving situations; through the use of physical models, drawings, calculators, and computers; 2. develop, test, and explain conjectures about properties of 2. read and write whole numbers and know place-value number systems and sets of numbers; and concepts and numeration through their relationships to counting, ordering, and grouping; 3. use number sense to estimate and justify the reasonableness of solutions to problems involving real 3. use numbers to count, to measure, to label, and to indicate numbers. location;

4. develop, test, and explain conjectures about properties of

5. use number sense to estimate and justify the

For students continuing their mathematics education reasonableness of solutions to problems involving whole beyond these standards, what they will know and are able numbers, and commonly-used fractions and decimals (for to do may include: example, 1/3, 3/4, 0.5, 0.75).

3. apply number theory concepts (for example, primes, factors,

4. use the relationships among fractions, decimals, and

percents, include the concepts of ratio and proportion, in problem-solving situations;

5. develop, test, and explain conjectures about properties of

integers and rational numbers; and

6. use number sense to estimate and justify the

reasonableness of solutions to problems involving integers, rational numbers, and common irrational numbers such as √ 2, √ 5, and π .

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Standard 2:Students use algebraic methods to explore, model, and describe patterns andfunctions involving numbers, shapes, data, and graphs in problem-solving situationsand communicate the reasoning used in solving these problems.

RATIONALE

The study of patterns, functions, and helps learners to recognize and generalize patterns; identify and clarify functionalrelationships; and represent and manipulate these relationships verbally, numerically, symbolically, and graphically.Symbolic representation, including the many interpretations of the concept of a variable, is important but only one ofmany ways to represent patterns and functions. Students who are adept at identifying and classifying patterns andfunctional relationships are better able to use these relationships in real situations, both in and out of school.

Because the understandings developed through this standard are critical to success in mathematics and to theappropriate use of quantitative reasoning in other disciplines, students should explore and use the ideas of functions,patterns, and algebra from kindergarten through 12th grade.

2. describe patterns and other relationships using tables, 2. represent functional relationships using written graphs, and open sentences; explanations, tables, equations, and graphs, and describing the connections among these representations; 3. recognize when a pattern exists and use that information to solve a problem; and 3. solve problems involving functional relationships using graphing calculators and/or computers as well as 4. observe and explain how a change in one quantity can appropriate paper-and-pencil techniques; produce a change in another (for example, the relationship between the number of bicycles and the numbers of wheels). 4. analyze and explain the behaviors, transformations, and general properties of types of equations and functions (for example, linear, quadratic, exponential); andGRADES 5-8 5. interpret algebraic equations and inequalities geometrically 1. represent, describe, and analyze patterns and relationships and describing geometric relationships algebraically. using tables, graphs, verbal rules, and standard algebraic notation;

2. describe patterns using variables, expressions, equations, For students continuing their mathematics education and inequalities in problem-solving situations; beyond these standards, what they know and are able to do may include: 3. analyze functional relationships to explain how a change in one quantity results in a change in another (for example, how • use rational, polynomial, trigonometric, and inverse the area of a circle changes as the radius increases, or how a functions to model real-world phenomena; person’s height changes over time); • represent and solve problems using linear programming 4. distinguish between linear and nonlinear functions through and difference equations; informal investigations; and • solve systems of linear equations using matrices and vectors; 5. solve simple linear equations in problem-solving situations • describe the concept of continuity of a function; using a variety of methods (informal, formal, graphical) and a • perform operations on and between functions; and variety of tools (physical materials, calculators, computers). • make the connections between trigonometric functions and polar coordinates, complex numbers, and series.

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Standard 3:Students use data collection and analysis, statistics, and probability in problem-solving situations and communicate the reasoning used in solving these problems.

RATIONALE

Statistics are used to understand how information is processed and translated into usable knowledge. Through the studyof statistics, students learn to collect, organize, and summarize data. In addition, statistics requires students to use datato ask and answer questions. Students also need to know how to analyze data and make decisions based on theirinterpretations. Probability extends statistical analysis to predicting the likelihood of future events and outcomes.Students learn probability — the study of chance — so that numerical data can be used to predict future events as wellas record the past.

BENCHMARKSGRADES K-4 GRADES 9-12

1. construct, read, and interpret displays of data including 1. design and conduct a statistical experiment to study a tables, charts, pictographs, and bar graphs; problem, and interpret and communicate the results using the appropriate technology (for example, graphing 2. interpret data using the concepts of largest, smallest, most calculators, computer software); often, and middle; 2. analyze statistical claims for erroneous conclusions or 3. generate, analyze, and make predictions based on data distortions; obtained from surveys and chance devices; and 3. fit curves to scatter plots, using informal methods or appropriate technology, to determine the strength of the 4. solve problems using various strategies for making relationship between two data sets and to make predictions; combinations (for example, determining the number of different outfits that can be made using two blouses and 4. draw conclusions about distributions of data based on three skirts). analysis of statistical summaries (for example, the combination of mean and standard deviation, andGRADES 5-8 differences between the mean and median); 1. read and construct displays of data using appropriate 5. use experimental and theoretical probability to represent techniques (for example, line graphs, circle graphs, scatter and solve problems involving uncertainty (for example, the plots, box plots, stem-and-leaf plots) and appropriate chance of playing professional sports if a student is a technology; successful high school athlete); and 2. display and use measures of central tendency, such as 6. solve real-world problems with informal use of combinations mean, median, and mode, and measures of variability, and permutations (for example, determining the number of such as range and quartiles; possible meals at a restaurant featuring a given number of side dishes). 3. evaluate arguments that are based on statistical claims;

4. formulate hypotheses, draw conclusions, and make For students continuing their mathematics education convincing arguments based on data analysis; beyond these standards, what they know and are able to do may include 5. determine probabilities through experiments or simulations;

6. make predictions and compare results using both

• create and interpret discrete and continuous probability distributions, and understand their application to real- experimental and theoretical probability drawn from real- world situations (for example, insurance); world problems; and • test hypotheses using appropriate statistics; 7. use counting strategies to determine all the possible • explore the effect of sample size on the results of outcomes from an experiment (for example, the number of statistical surveys using experiments and simulations; ways students can line up to have their picture taken). and • solve real-world problems with formal use of combinations and permutations.

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Standard 4:Students use geometric concepts, properties, and relationships in problem-solvingsituations and communicate the reasoning used in solving these problems.

RATIONALE

The process of recording and analyzing shapes and their properties became the branch of mathematics called geometry.Students who understand the concepts and language of geometry are better prepared to learn number andmeasurement ideas as well as other advanced mathematical topics. Students' spatial capabilities frequently exceed theirnumerical skills and tapping these strengths can foster an interest in mathematics and improve number understandingsand skills.

The goals of studying geometry include: understanding of shapes and of two- and three-dimensional relationships, howobjects are located in a plane or in space, symmetry and rotation, and visualization from different perspectives.

Encouraging students to make and test hypotheses about geometric concepts can begin in the primary grades.

3. relate geometric ideas to measurement and number sense; 3. make and test conjectures about geometric shapes and their properties, incorporating technology where appropriate; and 4. solve problems using geometric relationships and spatial reasoning (for example, using rectangular coordinates to 4. use trigonometric ratios in problem-solving situations (for locate objects, constructing models of three-dimensional example, finding the height of a building from a given point, objects); and if the distance to the building and the angle of elevation are known). 5. recognize geometry in their world (for example, in art and in nature).

GRADES 5-8

1. construct two- and three-dimensional models using a

variety of materials and tools; For students continuing their mathematics education beyond these standards, what they know and are able to do 2. describe, analyze, and reason informally about the may include: properties (for example, parallelism, perpendicularity, congruence) of two- and three-dimensional figures ; • deduce properties of figures using vectors; • apply transformations, coordinates, and vectors in 3. apply the concepts of ratio, proportion, and similarity in problem-solving situations; and problem-solving situations; • describe, analyze, and extend patterns produced by processes of geometric change (for example, limits and 4. solve problems using coordinate geometry; fractals).

5. solve problems involving perimeter and area in two

dimensions, and involving surface area and volume in three dimensions; and

6. transform geometric figures using reflections, translations,

and rotations to explore congruence.

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Standard 5:Students use a variety of tools and techniques to measure, apply the results inproblem-solving situations, and communicate the reasoning used in solving theseproblems.RATIONALE

Using agreed-upon units, such as inches, kilograms, heartbeats, paces, or degrees, we quantify the world in which welive. Measurement is one way to make numbers meaningful to students. Naturally, measurement is closely allied withgeometry (for example, through angular, linear, area, and volume measurements), but measurement involves more thanusing a ruler and a protractor. Measuring diverse quantities involves making connections within mathematics and acrossthe curriculum.

Students need to identify attributes they wish to measure and select the appropriate tools. Further, comparisons ofattributes, estimation and approximation allow students to apply measurement to solving problems.

2. compare and order objects according to measurable 2. select and use appropriate techniques and tools to attributes (for example, longest to shortest, lightest to measure quantities in order to achieve specified degrees heaviest); of precision, accuracy, and error (or tolerance) of measurements; and 3. demonstrate the process of measuring and explain the concepts related to units of measurement; 3. determine the degree of accuracy of a measurement (for example, by understanding and using significant digits). 4. use the approximate measures of familiar objects (for example, the width of your finger, the temperature of a 4. demonstrate the meanings of area under a curve and room, the weight of a gallon of milk) to develop a sense of length of an arc. measurement; and

5. select and use appropriate standard and non-standard

units of measurement in problem-solving situations.

GRADES 5-8 For students continuing their mathematics education 1. estimate, use, and describe measures of distance, beyond these standards, what they know and are able to do perimeter, area, volume, capacity, weight, mass, and may include: angle comparison; • demonstrate the meanings of area under a curve and 2. estimate, make, and use direct and indirect measurements length of an arc. to describe and make comparisons;

3. read and interpret various scales including those based on

number lines, graphs, and maps;

4. develop and use formulas and procedures to solve

problems involving measurement;

5. describe how a change in an object's linear dimensions

affects its perimeter, area, and volume; and

6. select and use appropriate units and tools to measure to

the degree of accuracy required in a particular problem- solving situation.

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Standard 6:Students link concepts and procedures as they develop and use computationaltechniques, including estimation, mental arithmetic, paper-and-pencil, calculators, andcomputers, in problem-solving situations and communicate the reasoning used insolving these problems.

RATIONALEComputation is an indispensable part of mathematics and our daily lives. We use it to balance our checkbooks, figureour taxes, and make business decisions. The basic facts of addition, subtraction, multiplication, and division are similarlyindispensable. Today's students must be able to effectively use a variety of computational tools and techniques includingestimation, mental arithmetic, paper-and-pencil, calculators, and computers. Estimation and mental arithmetic serve apractical function in our daily lives, and help students develop meaning for numbers and understanding of numberrelationships.

Computational skill is related to “operation sense”. Students build operation sense by modeling their understanding ofnumber operations and their properties, by describing how number operations are related to one another, and by seeinghow the use of a particular operation changes the value of the numbers involved.

BENCHMARKS GRADES K-4 GRADES 9-12

1. demonstrate conceptual meanings for the four basic 1. use ratios, proportions, and percents in problem-solving arithmetic operations of addition, subtraction, multiplication, situations; and division; 2. select and use appropriate algorithms for computing with 2. add and subtract commonly-used fractions and decimals real numbers in problem-solving situations and determine using physical models (for example, 1/3, 3/4, 0.5, 0.75); whether the results are reasonable; and

3. demonstrate fluency with basic addition, subtraction, 3. describe the limitations of estimation, and assess the multiplication, and division facts without the use of a amount of error resulting from estimation within acceptable calculator; limits.

4. construct, use, and explain procedures to compute and

estimate with whole numbers; and

5. select and use appropriate algorithms for computing with

whole numbers in problem-solving situations. For students continuing their mathematics education GRADES 5-8 beyond these standards, what they know and are able to do may include: 1. use models to explain how ratios, proportions, and percents can be used to solve real-world problems; • analyze and solve optimization problems; • analyze different algorithms (for example, sorting) for 2. construct, use, and explain procedures to compute and efficiency; estimate with whole numbers, fractions, decimals, and integers; • analyze and use critical path algorithms (for example, determining in which order to perform a set of tasks in a large project); and 3. develop, apply, and explain a variety of different estimation strategies in problem-solving situations, and explain why an • investigate problem situations that arise in connection estimate may be acceptable in place of an exact answer; with computer validation and the application of and algorithms.

4. select and use appropriate algorithms for computing with

commonly used fractions and decimals, percents, and integers in problem-solving and determine whether the results are reasonable.

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Colorado Model Content Standards for Mathematics

GLOSSARYAbsolute value — A number's distance from zero on a Exponent — A number used to tell how many times a number number line. The absolute value of -6, shown as |-6|, is 6, and or variable is used as a factor. For example, 53 indicates that the absolute value of 6, shown as |6|, is 6. 5 is a factor 3 times, that is, 5 x 5 x 5. The value of 53 is 125.Algebra — The branch of mathematics that is the Fractal — A geometric shape that is self-similar and has generalization of the ideas of arithmetic. fractional dimensions. Natural phenomena such as theAlgebraic methods — The use of symbols to represent formation of snowflakes, clouds, mountain ranges, and numbers and signs to represent their relationships. landscapes involve patterns. Their pictorial representations are fractals and are usually generated by computers.Algorithm — A step-by-step procedure. Function — A relationship between two sets of numbers (orBasic facts — Addition facts through 10 (0 + 0, 1 + 0, ... , 10 + other mathematical objects). Functions can be used to 10), subtraction facts which are the inverses of the addition understand how one quantity varies in relation to another, for facts (20 - 10, ... , 1 - 0, 0 - 0), multiplication facts (1 x 1, 1 x example, the relationship between the number of cars and 2, ... , 10 x 10), and division facts which are the inverses of the number of tires. the multiplication facts (1 ÷ 1, 2 ÷ 1, ... , 100 ÷ 10). Geometry — A branch of mathematics that deals with theBenchmark - Tactical description of the knowledge and skills measurement, properties, and relationships of points, lines, students should acquire within each grade level range (i.e., angles, and two- and three-dimensional figures. K-4, 5-8, 9-12). Integers — The set of numbers consisting of the countingBox plot (also called a box-and-whiskers plot) — A graphic numbers (that is, 1, 2, 3, 4, 5, ... ), their opposites (that is, method for showing a summary of data using median, negative numbers, -1, -2, -3, ... ), and zero. quartiles, and extremes of data. A box plot makes it easy to see where the data are spread out and where they are Irrational numbers — The set of numbers which cannot be concentrated. The longer the box, the more the data are represented as fractions. Examples are √2, 3√29, e , and π . spread out. Linear function — A function that has a constant rate ofCapacity — The volume of a container given in units of liquid change. measure. The standard units of capacity are the liter and the Logarithm — Alternate way to express an exponent. For gallon. example, log2 8=3 is equivalent to 23=8.Combinations — Subsets chosen from a larger set of objects Matrix (pl. matrices) — A rectangular array of numbers (or in which the order of the items doesn't matter (for example, letters) arranged in rows and columns. the number of different committees of three that can be Measures of central tendency — Numbers which in some chosen from a group of twelve members). sense communicate the "center" or "middle" of a set of data.Complex numbers — Numbers that can be written in the form The mean, median, and mode of statistical data are all a + bi, for example, -2.7 + 8.9i, where a and b are real measures of central tendency. numbers and i = √-1. Measures of variability — Numbers which describe howCongruent or the concept of congruence — Two figures are spread out a set of data is, for example, range and quartile. said to be congruent if they are the same size and shape. Mental arithmetic — Performing computations in one's headCoordinate geometry — Geometry based on the coordinate without writing anything down. Mental arithmetic strategies system. include finding pairs that add up to 10 or 100, doubling, andCoordinate system (also called rectangular coordinate halving. system) — A method of locating points in the plane or in Model — To make or construct a physical or mathematical space by means of numbers. A point in a plane can be representation. located by its distances from both a horizontal and a vertical Number sense — An understanding of number. This would line called the axes. The horizontal line is called the x-axis. include number meanings, number relationships, number The vertical line is called the y-axis. The pairs of numbers are size, and the relative effect of operations on numbers. called ordered pairs. The first number, called the x- coordinate, designates the distance along the horizontal axis. Open sentence — a statement that contains at least one The second number, called the y-coordinate, designates the unknown. It becomes true or false when a quantity is distance along the vertical axis. The point at which the two substituted for the unknown. For example, 3 + x = 5. axes intersect has the coordinates (0,0) and is called the Optimization problems — Real-world problems in which, origin. given a number of constraints, the best solution isConjecture — A statement that is to be shown true or false. A determined. For example, finding the best number of nonstop conjecture is usually developed by examining several specific flights from Denver to San Francisco given the cost of fuel, situations. number of passengers, number of crew required, etc.Dilation — A transformation that either enlarges or reduces a Patterns — Regularities in situations such as those in nature, geometric figure proportionally. events, shapes, designs, and sets of numbers (for example, spirals on pineapples, geometric designs in quilts, the number sequence 3,6,9,12,...).

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GLOSSARYPermutations — All possible arrangements of a given number Spatial visualization (also called spatial reasoning) — A type of of items in which the order of the items makes a difference. reasoning in which a person can draw upon one's understanding For example, the different ways that a set of four books can of relationships in space, the three-dimensional world. For be placed on a shelf. example, spatial reasoning is demonstrated by one's ability toPrime number — A counting number that can only be evenly build a three-dimensional model of a building shown in a picture. divided by two different numbers, 1 and the number itself. A person who uses spatial visualization is said to have spatial The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, sense. 29. Square root — That number which when multiplied by itself Probability — The likeliness or chance of an event occurring. produces the given number. For example, 5 is the square root of 25, because 5 x 5 = 25. Problem solving — Refer to the introduction (page Mathematics - 4). Problem-solving situations — Contexts in which problems Statistics — The branch of mathematics which is the study of the are presented that apply mathematics to practical situations methods of collecting and analyzing data. The data are collected in the real world, or problems that arise from the investigation on samples from various populations of people, animals, or of mathematical ideas. products. Statistics are used in many fields, such as biology, education, physics, psychology, and sociology. Quadratic function — A function that has an equation of the form y = ax2+bx+c, where a ≠ 0. These functions are used to Stem-and-leaf plot — A frequency distribution made by describe the flight of a ball and the stream of water from a arranging data. It is one way of visually portraying data that is fountain. frequently used in newspapers and magazines because it provides an efficient way of showing information as well asRational numbers — A number that can be expressed in the comparing different sets of data. form a/b, where a and b are integers and b ≠ 0, for example, 3/4, 2/1, or 11/3. Every integer is a rational number, since it can be Symmetry — The correspondence in size, form, and expressed in the form a/b, for example, 5 = 5/1. Rational arrangement of parts on opposite sides of a plane, line, or point. numbers may be expressed as fractional or decimal numbers, for For example, a figure that has line symmetry has two halves example, 3/4 or .75. Finite decimals, repeating decimals, and which coincide if folded along its line of symmetry. mixed numbers all represent rational numbers. Transformation — The process of changing one configuration orReal numbers — All rational and irrational numbers. expression into another in accordance with a rule. Common geometric transformations include translations, rotations, andReal-world problems (also called real-world experiences) — reflections. Quantitative problems that arise from a wide variety of human experiences which may take into consideration contributions Translation (also called a slide) — A transformation that moves a from various cultures (for example, Mayan or American geometric figure by sliding. Each of the points of the geometric pioneers), problems from abstract mathematics, and applications figure moves the same distance in the same direction. to various careers (for example, making change or calculating Trigonometric ratios — The ratios of the lengths of pairs of sides the sale price of an item). in a right triangle. There are three basic trigonometric ratios usedReflection (also called a flip) — A transformation which produces in trigonometry: sine (sin), cosine (cos), and tangent (tan). the mirror image of a geometric figure. Trigonometry — A branch of mathematics that combinesRotation (also called a turn) — A transformation which turns a arithmetic, algebra, and geometry. Trigonometry is used in figure about a point a given number of degrees. surveying, navigation, and various sciences such as physics.Scatter plots (also called scatter diagram or scattergram) — A Variable — A quantity that may assume any one of a set of graph of the points representing a collection of data. values. In the equation 2x + y = 9, x and y are variables.Scientific notation — A short-hand way of writing very large or Vector — A quantity which has both magnitude and direction. very small numbers. A number expressed in scientific notation is Vectors may be interpreted as physical quantities such as expressed as a decimal number between 1 and 10 multiplied by velocity and force. a power of 10, for example, 4.53 X 103 = 4350. Volume — The measure of the interior of a three-dimensionalSimilarity — Objects or figures that are the same shape are figure. A unit for measuring volume is the cubic unit. similar figures. They are not necessarily the same size. If two figures are similar, we say that there is similarity between the figures.